Unit 5: Continuous Probability Distributions

5.1 Continuous Uniform Distribution

• Story: Any value within a given interval [a, b] is equally likely. It represents a complete lack of knowledge about the value, except that it's in the interval.
• Example: A random number generator that produces a number between 0 and 1.
• Parameters: a (lower bound), b (upper bound). We write X ~ U(a, b).
• P.D.F.:
  f(x) = 1 / (b - a), for a ≤ x ≤ b

  (And f(x)=0 otherwise. The graph is a simple rectangle of height 1/(b-a) and width (b-a), so the area is 1).

• Mean: E[X] = (a + b) / 2
• Variance: Var(X) = (b - a)² / 12
• M.G.F.: M(t) = (e^tb - e^ta) / (t * (b - a))

5.2 Normal Distribution (Gaussian Distribution)

• Story: The most important distribution in all of statistics. It's the "bell curve." It arises naturally from the Central Limit Theorem, which states that the sum of many independent random variables tends to be normally distributed. It's used to model phenomena like height, IQ scores, measurement errors, etc.
• Parameters: μ (mu, the mean), σ² (sigma-squared, the variance). We write X ~ N(μ, σ²).
• P.D.F.:
  f(x) = [ 1 / (σ * sqrt(2π)) ] * e^{-(x - μ)² / (2σ²)}, for -∞ < x < +∞
• Mean: E[X] = μ
• Variance: Var(X) = σ²
• M.G.F.: M(t) = e^{(μt + σ²t²/2)}
• Property (Additivity): If X ~ N(μ₁, σ₁²) and Y ~ N(μ₂, σ₂²) are independent, then (aX + bY) is also Normal.
  Specifically, (X+Y) ~ N(μ₁ + μ₂, σ₁² + σ₂²).

Standard Normal Distribution (Z)

• This is a special case of the Normal distribution with μ = 0 and σ² = 1. We write Z ~ N(0, 1).
• P.D.F.: φ(z) = [ 1 / sqrt(2π) ] * e^-z²/2
• Standardization: Any Normal variable X ~ N(μ, σ²) can be transformed into a Standard Normal variable Z by:
Z = (X - μ) / σ

This is how we use Z-tables to find probabilities for any Normal distribution.

5.3 Exponential Distribution

• Story: Describes the waiting time (X) until the next event in a Poisson process.
• Examples:
  • Time between phone calls at a call center.
  • Time until a radioactive particle decays.
  • Lifetime of a component that fails at a constant rate.
• Parameters: λ (lambda, the "rate" of events; same λ as in the Poisson process).
• P.D.F.:
  f(x) = λ * e^-λx, for x ≥ 0
• Mean: E[X] = 1 / λ (If the rate is 5 calls/hour, the mean wait time is 1/5 = 0.2 hours).
• Variance: Var(X) = 1 / λ²
• M.G.F.: M(t) = λ / (λ - t), for t < λ
• Property (Memorylessness): Just like the Geometric, the Exponential is memoryless. P(X > s+t | X > s) = P(X > t). If a lightbulb has worked for 100 hours, its probability of working for another 50 hours is the same as a brand new bulb working for 50 hours.

5.4 Gamma Distribution

• Story: A generalization of the Exponential. It is the waiting time (X) until α (alpha) events have occurred in a Poisson process.
• Parameters: α (alpha, the "shape" parameter, or number of events), λ (lambda, the "rate" parameter). We write X ~ Gamma(α, λ).
• P.D.F.:
  f(x) = [ λ^α / Γ(α) ] * x^α-1 * e^-λx, for x ≥ 0

  Where Γ(α) (Gamma function) is a generalization of the factorial. Γ(n) = (n-1)! for integers. Γ(α) = ∫₀^∞ t^α-1e^-t dt.

• Mean: E[X] = α / λ
• Variance: Var(X) = α / λ²
• M.G.F.: M(t) = (λ / (λ - t))^α
• Relationships:
  • Gamma(1, λ) = Exponential(λ). (Set α=1 in all formulas).
  • The sum of α independent Exponential(λ) variables is a Gamma(α, λ) variable.
  • The Chi-Square (χ²) distribution is a special case of the Gamma.

5.5 Beta Distribution

• Story: Very flexible distribution used to model variables that are constrained to be between 0 and 1, such as proportions, percentages, or probabilities.
• Examples:
  • The proportion of defective items in a shipment.
  • The fraction of time a machine is operational.
• Parameters: α (alpha, shape), β (beta, shape). We write X ~ Beta(α, β).
• P.D.F.:
  f(x) = [ 1 / B(α, β) ] * x^α-1 * (1-x)^β-1, for 0 ≤ x ≤ 1

  Where B(α, β) (Beta function) = Γ(α)Γ(β) / Γ(α+β).

• Mean: E[X] = α / (α + β)
• Variance: Var(X) = (αβ) / [ (α+β)² * (α+β+1) ]
• Note: If α=1, β=1, then f(x)=1. This is the Uniform(0, 1) distribution.

5.6 Cauchy Distribution

• Story: A "pathological" distribution famous for being a counterexample. It's the ratio of two independent Standard Normal (N(0,1)) variables.
• Example: A lighthouse at (0, 1) spins. The x-coordinate where the light beam hits the x-axis follows a Cauchy distribution.
• Parameters: θ (theta, location/median), γ (gamma, scale).
• P.D.F.:
  f(x) = 1 / [ πγ * (1 + ((x-θ)/γ)²) ]
• Extremely Important Property: The Mean (E[X]) and Variance (Var(X)) are UNDEFINED.

  The integrals ∫ x*f(x) dx and ∫ x²*f(x) dx do not converge. The M.G.F. also does not exist. The "tails" of the distribution are too "fat."

• Median: θ

5.7 Weibull Distribution

• Story: Widely used in reliability and survival analysis to model the "time to failure" of a component. It is more flexible than the Exponential because its failure rate can change over time.
• Parameters: k (shape), λ (lambda, scale).
• P.D.F.:
  f(x) = (k/λ) * (x/λ)^k-1 * e^-(x/λ)k, for x ≥ 0
• Relationship to Exponential: If k=1 (shape parameter is 1), the Weibull distribution simplifies to the Exponential(1/λ) distribution. (Note the parameter inversion).
• Mean: E[X] = λ * Γ(1 + 1/k)
• Variance: Var(X) = λ² * [ Γ(1 + 2/k) - (Γ(1 + 1/k))² ]

5.8 Laplace Distribution (Double Exponential)

• Story: Looks like two Exponential distributions placed back-to-back. It is "peaked" at the mean and has fatter tails than a Normal distribution, making it useful for modeling data with more outliers.
• Parameters: μ (location/mean), b (scale).
• P.D.F.:
  f(x) = [ 1 / (2b) ] * e^{-|x - μ| / b}, for -∞ < x < +∞
• Mean: E[X] = μ
• Variance: Var(X) = 2b²

5.9 Summary Table & Relationships